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logic-proofs/03-axioms-and-godel.md

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+# Axioms and Gödel's Incompleteness Theorem
+
+**Source:** MIT 6.1200J Lecture 01
+
+## Axiom
+**Definition:** A proposition that is assumed to be True.
+
+### Key Insight
+- You MUST make assumptions in mathematics
+- The key is to **state them upfront**
+
+## Examples of Contradictory Axiom Systems
+All three are equally valid — they define different geometric worlds:
+
+1. **Euclidean:** Given line l and point p ∉ l, exactly one line through p parallel to l
+2. **Hyperbolic:** Given line l and point p ∉ l, infinitely many lines through p parallel to l
+3. **Spherical:** Given line l and point p ∉ l, no line through p parallel to l
+
+**Lesson:** Different axioms yield different proofs and theorems. Anyone who accepts your axioms must accept theorems derived from them.
+
+## Consistency and Completeness
+
+### Consistent
+Definition: A set of axioms is consistent if no proposition can be both proved and disproved.
+
+### Complete
+Definition: A set of axioms is complete if every proposition can be either proved or disproved.
+
+**Ideal scenario:** Both consistent AND complete.
+
+## Gödel's Incompleteness Theorem
+
+**Theorem (Kurt Gödel, 1930s):** No set of axioms is both complete and consistent.
+
+### Impact
+- Shocked the mathematical community
+- Logicians Russell and Whitehead spent careers trying to find complete + consistent axioms for arithmetic — impossible!
+
+### Corollary
+If axioms must be consistent (necessary for validity), then there exist **True statements that cannot be proved**.
+
+### Example
+- Goldbach's Conjecture might be true but unprovable
+- (If so, please don't assign it as homework!)
+
+## Related
+- [[02-propositions|Propositions]]
+- [[01-what-is-a-proof|What is a Proof?]]